Note that we have both “strangle,” the strangler class feature, and “Strangle,” the feat, here. I will try to be clear which I am referring to but in the interests of trying to keep this long answer from getting longer, I will sometimes rely on that capital to distinguish the two.
The rules here are an absolute mess. We have several questions that the rules do not adequately answer:
When strangle says that the target is “considered flanked for the purpose of using this ability,” does that mean that when using this ability, the target is considered flanked in general, or only for this ability?
Is the damage from the Strangle feat a retroactive bonus on the damage already dealt, or is it a separate instance of damage?
In the case of non-stacking rolled bonuses, can you roll each instance and choose the highest result?
Depending on the answers to these questions, the damage bonus could be
+2d6, if Strangle is a bonus and none of these bonuses stack.
+2d6 rolled two or three times, taking the highest result, in the above case where overlapping bonuses are allowed to be rolled repeatedly. The difference between two and three rerolls is whether or not the flanking during strangle allows the rogue sneak attack dice to come into play.
+3d6, if Strangle is a bonus that doesn’t stack, and strangle just stacks with rogue sneak attack and prevents Accomplished Sneak Attacker from applying.
+4d6, if strangle doesn’t trigger rogue sneak attack at all, but Strangle is a separate instance of damage so Accomplished Sneak Attacker applies. Becomes +5d6 or +6d6 if strangle is used against a foe that is actually flanked or lacking Dex to AC.
+4d6, if Strangle is a bonus that doesn’t stack, but strangle applies rogue sneak attack that stacks without blocking Accomplished Sneak Attacker. Arguably you could actually get +2d6+(2d6 rolled twice taking the best result).
+5d6, if Strangle is separate, and both strangle and rogue sneak attack apply (either because strangle does it or because the target just actually is flanked or lacking Dex to AC), and that stacks, but blocks Accomplished Sneak Attacker in the process.
+6d6, if Strangle is separate, and both strangle and rogue sneak attack apply (either because strangle does it or because the target just actually is flanked or lacking Dex to AC), and that stacks without blocking Accomplished Sneak Attacker.
Not all of these results are equally-likely under the rules. There are reasons to suspect that one answer to a given question is more likely than another. I will go through my reasoning for each below, but my conclusion is that the most likely result is actually the +6d6. The next-most-likely result is +4d6, on the basis that strangle doesn’t trigger rogue sneak attack at all, but in this case it would go up to +6d6 if used on someone that would trigger sneak attack independently of strangle.
From a balance perspective, a fairly-normal rogue is looking at getting +2d6 twice at this level (+4d6 total) thanks to two-weapon fighting. Thus +6d6, on top of grappling, is arguably better. On the other hand, rogues will scale to more attacks (even if you get greater grapple and rapid grapple, that maxes at three instances of your damage), and a dual-wielder could also manage +3d6 with judicious multiclassing—which, of course, also has drawbacks. The +6d6, for two feats, multiclassing, and lacking growth potential... seems OK to me. The +4d6 is certainly OK, matching what a dual-wielding rogue does for less investment; in fact, I would call this underpowered under that interpretation, though the ability to get that damage “back” by using strangle on a flanked or missing-Dex foe helps.
So, how do we answer these questions, to come to those conclusions?
The first thing to address is whether or not the strange class feature triggers the rogue sneak attack. Personally, I prefer to read “for the purpose of using this ability” here as meaning “when using this ability,” but that it does still trigger rogue sneak attack. After all, you are dealing damage to a creature you treat as flanked while using this ability—that is the trigger for sneak attack. But it is also definitely arguable that the creature doesn’t count as flanked for the purposes of the rogue sneak attack.
Note that even if you limit strangle in this fashion, you could always use strangle against a foe that is actually flanked or has lost his Dexterity bonus to AC, so you can definitely still trigger the rogue sneak attack damage here even if it takes a little more work.
The next issue is, Accomplished Sneak Attacker adds 1d6 damage to sneak attack, but only if you are high enough level for that to not exceed half your level rounded up. You are 3rd level, so half rounded up is a max of 2d6, and your rogue sneak attack damage is 1d6, so that should be fine, but you also can get sneak attack damage from your strangler class features, another 2d6 for a total of 3d6. It is unclear whether or not that should “count” towards the limit on Accomplished Sneak Attacker. However, even more complicated issues, discussed below, suggest that the class features do not directly stack together but remain separate. Thus, rolling 2d6 for rogue sneak attack damage seems legitimate.
Now then, strangle states that you are considered flanking for the purposes of dealing sneak attack damage to the target when you deal damage or pin. This implies that both the sneak attack from the strangler archetype and the sneak attack from rogue would apply. However, neither rogue nor strangler addresses stacking. Most sources of sneak attack do, but rogue, as the “original” source, doesn’t, and neither does strangler, presumably because it’s so odd. So if you don’t just combine your dice together (which I don’t think we do, and this is a good thing for the purposes of Accomplished Sneak Attacker), but rather have two separate bonuses from two separate class features, do those stack?
RAW, probably. They’re both untyped bonuses (“sneak attack” is not a bonus type), and they come from different sources (strangle vs. sneak attack, respectively), but since strangle calls the damage out as specifically sneak attack damage, you could argue that it counts as the sneak attack class feature for the purposes of determining the source, which means... well maybe they still stack, because it’s not at all clear that two separate instances of the same class feature count as the “same source” per the bonus-stacking rules. But you could argue they do, maybe.
So now we use a grapple to deal damage, and strangle means we count as flanking and also have a +2d6 damage bonus. Rogue’s sneak attack with Accomplished Sneak Attacker also triggers, since we are dealing damage to flanked opponent, and so that’s another +2d6 damage. These may or may not stack together (but RAW, probably do), so that’s either a total of +4d6 damage, or else you roll 2d6 twice and add the higher result.
After that we use the Strangler feat as a swift action to deal our sneak attack damage again. Whether this is in addition to the fact that we already are or not is unclear; as a separate action, though, I would argue this is a separate source of damage. The strangle sneak attack damage probably does not apply here, but the rogue sneak attack damage does. Thus we deal a separate instance of 2d6 damage. Note that this would be subject to separate damage reduction, which may be a concern.
If instead you rule that the Strangler feat is adding a bonus to our already-existing damage, rather than a separate instance of damage, then we definitely run into stacking issues: our rogue sneak attack damage already applies, and attempting to do so again would result in the same untyped bonus from the same source, and not stack. At best, you would get to roll 2d6 again to choose the higher result.